A bicycle odometer (which measures distance traveled) is attached near the wheei6h 94huja /phykaupof k-aq ;-8 g+s:l hub and is designed for 27-inch wheels. What hakpg4q +k8afa /puui69:h- yh -;as ojhppens if you use it on a bicycle with 24-inch wheels?

Suppose a disk rotates at constant angularm*g fmo qic;;em-fn ;. velocity. Does a point on the rim have radial and/or tangential acceleration? If the disk’s angular velocity increases uniformly;fqc* m;ne;mm o. ifg-, does the point have radial and/or tangential acceleration? For which cases would the magnitude of either component of linear acceleration change?

Could a nonrigid body be described by a single value of the angulaqi:/o1uijno,98ba 3wjlm q* br velocity $\omega$ Explain.

Can a small force ever exert ayut- jicfu .sad0: m85 greater torque than a larger force? Explain.

If a force $\vec{F}$ acts on an object such that its lever arm is zero, does it have any effect on the object’s motion? Explain.

Why is it more difficult to do a sit-up with your hands q ids.rcq*g;d:qxz .y l.(vd/tj7*m rbehind your head than when your arms are stretched oudtrd*. s7d/q q zqi(m. *y g.:x;lvcjrt in front of you? A diagram may help you to answer this.

A 21-speed bicycle has seven sprockets at the rear wheel and three o6dacui ,(cy+*4 dbj83n-s w jcjda8nat the pedal cranks. In which gear is it harder to pedal, a small rear sprocket or a large rear sprocket? Why? In which gear is it harder to pedal, a small front sprocket or a large froni-3d jj6 8 jbacc4n*s +anod(cu 8dw,yt sprocket? Why?

Mammals that depend on being30dhz ammry8* able to run fast have slender lower legs with flesh and muscle concentrated high, close to the body (Fig. 8–34). On r3mdh0yazm 8* the basis of rotational dynamics, explain why this distribution of mass is advantageous.

Why do tightrope walkers4q) 8sgqt f2os3:v y24dklmdt (Fig. 8–35) carry a long, narrow beam?

If the net force on a system is zero, is the net torque also zero0e1b/o 2v s )/gwcwuqf? If the net torque on a system is zero, is tsu 20qbvw o)gw/f1c /ehe net force zero?

Two inclines have the same height but make different anglei3 mr9o ey9c7us with the horizontal. The same steel ball is rolled down each incline. On which incline will the speed ofre9y i9 c7muo3 the ball at the bottom be greater? Explain.

Two solid spheres simultaneously start rollinnxi v)4r 8dc/fac3v4pg (from rest) down an incline. One sphere has twice the radius8) /vv ai4 cn34pfxdcr and twice the mass of the other. Which reaches the bottom of the incline first? Which has the greater speed there? Which has the greater total kinetic energy at the bottom?

A sphere and a cylinder have the same rad;ub/qysacxh5:h 1 hf*ius and the same mass. They start from rest at the top of an incline. Which reaches the bottom first? Which has the greater speed at:fq5;yxsa/uh 1 hhb*c the bottom? Which has the greater total kinetic energy at the bottom? Which has the greater rotational KE?

We claim that momentum and angular 2,;myh 0bhgca7-o cxdxk 1h6vmomentum are conserved. Yet most moving or rotating objects eventually slow down and stop. Exp x 0dvh-c2k1ag,o6hh7mxc ;by lain.

If there were a great migration of people toward the Earth’s e+,xe*za3 sw6izex 9c rquator, how would this affect the length of the ixxza+3 z , cwse*6er9day?

Can the diver of Fig. 8–29 do a s/5k fxkj8pm,o omersault without having any initial rotation when she leaves the b j5kxm 8,ofp/koard?

The moment of inertia of a rotating solid disk about an axis thfb)ec/; x5:js d9s jahrough its center of mass ie);b9j:/saf x hsdc5js $\frac{1}{2}WR^2$ (Fig. 8–21c). Suppose instead that the axis of rotation passes through a point on the edge of the disk. Will the moment of inertia be the same, larger, or smaller?

Suppose you are sitting on a rotating stool holding a 2-kg mass in each nhdm)39lqjxhti h*9/ 4kk,uez a1x,t outstretched hand. If you suddenly drop the masses, will your angular velocity increase, decrease, or stay the s3kqta9en h,tu,dkxl )9xhhjz1/m 4*iame? Explain.

Two spheres look identicalh6 h5exzt 1e8b and have the same mass. However, one is hollow and the other is soli51z86h theex bd. Describe an experiment to determine which is which.

The angular velocity of a wheel rotating on a horiz gqi) /mchxa/q/1lh :utfv qi63i*qb:ontal axle points west. In what direction is the linear velocity of a point on the top of the wheel? If the angular acceleration points east, descrcil/ /gt*fiqmu6aih1:vq/xq) bhq 3:ibe the tangential linear acceleration of this point at the top of the wheel. Is the angular speed increasing or decreasing?

Suppose you are standing on the edge of a large vc+yc q .+n/ b*o+2ve xyu6t0u4txortfreely rotating turntable. What x*yu etct+o.n v ry2c6 v0t4b+x/uq+ohappens if you walk toward the center?

A shortstop may leap int /onpqq:gs 6)xrk.u r1-b4io io the air to catch a ball and throw it quickly. As he throws the ball, the upper part of his body rotates. If you look quickly you will notice that his hips and legs rotate in the opposite direcuqbro/i.):s1rkx-nq 4 g ipo6tion (Fig. 8–36). Explain.

On the basis of the law of conservation of angular momentum, discus ,qi 7jj5o5g)3mpt jjss why a helicopter must have more than one rotor (or propeller). Discuss one or more ways the second propeller can operate to keep the helicopter p5m)otjj7 s3ig 5j,jqstable.

  Express the following angles in radians: (a) 30nop7c4nejyf)uvnd*e: 48jz jn v: 2 :n $^{\circ} $, (b) 57 $^{\circ} $, (c) 90 $^{\circ} $, (d) 360 $^{\circ} $, and (e) 420 $^{\circ} $. Give as numerical values and as fractions of $\pi$.(Round to two decimal places)
(a)   $rad$ (b)   $rad$ (c)    $rad$ (d)    $rad$ (e)    $rad$

  Eclipses happen on Earth because of an amazing coincidence. Calculate, using thys(ae :o 8(ifdbc3v 0be informi svb(0d f8b(3oy cea:ation inside the Front Cover, the angular diameters (in radians) of the Sun and the Moon, as seen on Earth.
Sun =    $rad$ Moon =    $rad$

  A laser beam is directed at thjpp:52vc7 fb )qqww )ichf e+14 u,itge Moon, 380,000 km from Earth. The beam diverges at an a,:uqt54bwphgfp)iv wc1 2jcf q+ )ie7ngle $\theta$ (Fig. 8–37) of $1.4\times10^{-5}$ rad What diameter spot will it make on the Moon?    m

  The blades in a blender rotate at a rab 2:52)v(j u -gb6 bxzpaxfptnte of 6500 rpm. When the motor is turned off during operation, the blades slow to rest in 3.0 s. Wha5-j pp:bf2xt u)gbxz(ba26 nvt is the angular acceleration as the blades slow down?    $rad/s^2$

  A child rolls a ball on a 0bprz :-7yypw8c w0p d)b2f62qo cvrslevel floor 3.5 m to another child. If the ball makes 15.0 revolutions,wpb2crdr2sc-:wpo fpyz)7 0b0 v8 q6 y what is its diameter?    m

  A bicycle with tires 68 cm in diameter travels 8.0 km. How many rebo -kw79)6ufd a* ofkevolutions do thd9oo)k6 u ebk awf7f*-e wheels make?    $rev$

  (a) A grinding wheel 0.35 m in diameter rotat/r wj5pzc1 r6b oi8dmva )78yyes at 2500 rpm. Calculate its angular velocitz1rbrodci)pvm y76y j85 /aw8y in $rad/s$ $\omega$ =    $rad/sec$
(b) What are the linear speed and acceleration of a point on the edge of the grinding wheel? v =    $m/s$ $a_R$ =    $ m/s^2$

  A rotating merry-go-round makesl gk fb7.; 9w8t0yzpzcu9p df, one complete revolution in 4.0 s (Fig. 8–38). (a) Wha,uc7pkgfy; d98 . 9zz 0wtlbpft is the linear speed of a child seated 1.2 m from the center?    $m/s$
(b) What is her acceleration (give components)?    $m/s^2$  ----待选择----towardsaways  the center

  Calculate the angula 0nqcalqm i)xh4uw75g.xz8c /r velocity of the Earth (a) in its orbit around the Sun    $ \times10^{-7 }$ $rad/s$
(b) about its axis.    $ \times10^{-5}$ $rad/s$

  What is the linear speed of a poinsancw sp.7*o +*cze x*y , 5mzup1(sknt
(a) on the equator,    $m/s$
(b) on the Arctic Circle (latitude 66.5$^{\circ} $ N),    $m/s$
(c) at a latitude of 45.0$^{\circ} $ N, due to the Earth’s rotation?    $m/s$

  How fast (in rpm) must a centrifuge rotate if a particle 7.ao.r qmdn)wgd(hv 1 zpbl)da*/ -x26z 0 cm from the axis of rotation is to experience an acceleration of 1abd- ) m/n*w(p6vrgzo)xz hd 2ladq1 .00,000 $g’s$?    $rpm$

  A 70-cm-diameter wheel accelerates uniformly about iao oai:6j w2:pzuhca, * zhk1,ts center from 130 rpm to 280 rpm in 4.0 s. Determ:*z2, hoapac oiu,w:za 1k6 hjine
(a) its angular acceleration,$\approx$    $rad/s^2$(Round to one decimal places)
(b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating. $a_R$    $m/s^2$ $a_{tan}$    $m/s^2$

A turntable of radius )4w8 v: rfzvfnu y;pb-m+e0pm$R_1$ is turned by a circular rubber roller of radius $R_2$ in contact with it at their outer edges. What is the ratio of their angular velocities, $\omega_1$ / $\omega_2$

  In traveling to the Moon, astronauts aboard the Apollo spacecraft put themselvea6i9uj: ka 5des into a slow rotation to distribute the Sun’s energy evenly. At the start of their trip, they accelerated from no rotation to 1.0 revolution every minute during a kj:9u6da ia5e12-min time interval. The spacecraft can be thought of as a cylinder with a diameter of 8.5 m. Determine
(a) the angular acceleration, $\approx$    $rad/s^2$
(b) the radial and tangential components of the linear acceleration of a point on the skin of the ship 5.0 min after it started this acceleration. $a_{tan}$ =    $ \times10^{ -4}$ $m/s^2$ $a_{rad}$ =    $ \times10^{ -3}$ $m/s^2$

  A centrifuge accelerates uniformly from rw 6.sk;ovm 7 0tjqv.yiest to 15,000 rpm in 220 s. Through how many.ivw;s.j0vo7y kt 6q m revolutions did it turn in this time?    $rev$

  An automobile engine slows down from 4500 rpm to 1200 rpm in 2.s8elpp c7ujt: (as n385 s. Calculate
(a) its angular acceleration, assumed constant,    $rad/s^2$
(b) the total number of revolutions the engine makes in this time.    $rev$

  Pilots can be tested forfqg5r( *ga vk9 the stresses of flying highspeed jets in a whirling “human centrifuge,” which takes 1.0 min to turn through 20 complete revolutions before reaching its final s* gga59(rqkv fpeed.
(a) What was its angular acceleration (assumed constant),    $rev/min^2$
(b) what was its final angular speed in rpm?    $rpm$

  A wheel 33 cm in diameter accelerates uniformly from 240 rpm to 360 rpm in 6.5 rfeb84vkp i,( (7uuigjrj pwak),) ;zs. How far will a point on the edge u4;ep(g )f i,v(ba ji8k,w7 pkrzuj )rof the wheel have traveled in this time?    m

  A cooling fan is turned off when it is running at 850rev/min It turns 1500 -m8;ej )afw4f5ja5oan euwr2 revolutions befo2aa mf4 8a5we wfj;5e)rn juo-re it comes to a stop.
(a) What was the fan’s angular acceleration, assumed constant?    $\frac{rad}{s^2}$
(b) How long did it take the fan to come to a complete stop?    s

  The tires of a car make 65 revow zphgia(.f 9*lutions as the car reduces its speed uniformly from 95km/h to 45km/h The tires a*. 9pz w(ifhghave a diameter of 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  The tires of a car make 65 revolutions as the car reduces itb 1x1/7x7igq bh,mopys speed uniformly from 95km/h to 45km/h The tires have a diameter ogb 1x 7/ o1qbhpx,i7ymf 0.80 m.
(a) What was the angular acceleration of the tires? $\approx$    $rad/s^2$
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s

  A 55-kg person riding a bike puts all her weight onah2mrw*4ni 5 02 qjhhf each pedal when climbing a hill. The pedals rotate in a circle of radius 1i*rmnh 0q2 a5 wjh4h2f7 cm.
(a) What is the maximum torque she exerts?    $m \cdot N$
(b) How could she exert more torque?

  A person exerts a force of 55 N on the end of a dooruodw 73fieq 4 8q8v2 ttm79epz 74 cm wide. What is the magnitude of thvue 3728moe4pt 9q8dit7 zfw qe torque if the force is exerted
(a) perpendicular to the door    $m \cdot N$
(b) at a 45 $^{\circ} $ angle to the face of the door?    $m \cdot N$

  Calculate the net torque about the axle of the wheel shown in Fig. 8–3l6 3lhws dw6kq,,nb a, 6c6hpee:h(c n9. Assume that q ,, 6cwlda6sek3nh, 6: l(6bcwhepnha friction torque of 0.4 $m \cdot N$ opposes the motion.    $m \cdot N$  ----待选择----“counterclockwiseclockwise 

Two blocks, each of mass m, are attached to the ends of a massless rod whf yzl205/f :aleyvj;pb)w8 sj ich pivots as shown in Fwlyj/0 p5zev8slf b2af;j) : yig. 8–40. Initially the rod is held in the horizontal position and then released. Calculate the magnitude and direction of the net torque on this system.

  The bolts on the cylinder head of an dh vs4axp m19:engine require tightening to a torque of 38m 9p:hxvsa 14d $m \cdot N$ If a wrench is 28 cm long, what force perpendicular to the wrench must the mechanic exert at its end?    N
If the six-sided bolt head is 15 mm in diameter, estimate the force applied near each of the six points by a socket wrench (Fig. 8–41).    N

  Determine the moment of inertia of a *bg lo:l *u3/)wb jfpd30me jd10.8-kg sphere of radius 0.648 m when the axis of rotation is through its center. 3jl*0b oj p):3ldfugmbw/e d*    $kg \cdot m^2$

  Calculate the moment of inertia of a bicythcuc8s x 545 nzzdt/4cle wheel 66.7 cm in diameter. The rim and tire have a combined mass of 1.25 kg. The mass of the hub cu4 xnc548t z /zdchts5an be ignored (why?).    $kg \cdot m^2$

  A small 650-gram ball f) 7 ,p1luhh:4ga*z9ur*rlrgh2 tnmbon the end of a thin, light rod is rotated in a horizontal circle of radius 1.2mf u natr*h4h*:ug92p,b g) z l71rrhl m. Calculate
(a) the moment of inertia of the ball about the center of the circle,    $kg \cdot m^2$
(b) the torque needed to keep the ball rotating at constant angular velocity if air resistance exerts a force of 0.020 N on the ball. Ignore the rod’s moment of inertia and air resistance.    $m \cdot N$

  A potter is shaping a bowl on a potter’s wheel rotating at con9iq7do:oogrud4 8 r0wstant angular speed (Fig. o 9 do:rd7i08u wg4oqr8–42). The friction force between her hands and the clay is 1.5 N total.
(a) How large is her torque on the wheel, if the diameter of the bowl is 12 cm?    $m \cdot N$
(b) How long would it take for the potter’s wheel to stop if the only torque acting on it is due to the potter’s hand? The initial angular velocity of the wheel is 1.6 rev/s, and the moment of inertia of the wheel and the bowl is 0.11 $kg \cdot m^2$.    s

  Calculate the moment of insns m3zy6la, pb6l9-qertia of the array of point objects shown in Fig. 8–43 abouqylp b3n-slm 6,sz69a t
(a) the vertical axis,    $kg \cdot m^2$
(b) the horizontal axis. Assume m=1.8 kg,M=3.1kg and the objects are wired together by very light, rigid pieces of wire. The array is rectangular and is split through the middle by the horizontal axis.    $kg \cdot m^2$
(c) About which axis would it be harder to accelerate this array?

  An oxygen molecule consists of two oxygen atoms whos f -pi4kkfw7dgm 8e;1we total mass is $5.3 \times10^{ -26}$ kg and whose moment of inertia about an axis perpendicular to the line joining the two atoms, midway between them, is $ 1.9\times10^{-46 }$ $kg \cdot m^2$ From these data, estimate the effective distance between the atoms.    $\times10^{-10 }$ m

  To get a flat, uniform cylindrical satellite spinnuk: tq;p6mj e6ing at the correct rate, engineers fire four tangential rockets as shown in Fig. 8–44. If the satellite has a mass of 3600 kg and a radius of 4.0 m, what is the r ju:qtkp6e 6;mequired steady force of each rocket if the satellite is to reach 32 rpm in 5.0 min? $\approx$    N(round to the nearest integer)

  A grinding wheel is a uniform cylinder with a radius of 8.5i.* upswboa1/6qmj8 fg o7t5c0 cm and a mass of 0.580 kg. Calculato*q/1b5 pg.us6owt87mci jf ae
(a) its moment of inertia about its center, $\approx$    $kg \cdot m^2$
(b) the applied torque needed to accelerate it from rest to 1500 rpm in 5.00 s if it is known to slow down from 1500 rpm to rest in 55.0 s。    $m \cdot N$

  A softball player swings a bate )k7(( 7j/qqlsgie gs, accelerating it from rest to 3 $rev/s$ in a time of 0.20 s. Approximate the bat as a 2.2-kg uniform rod of length 0.95 m, and compute the torque the player applies to one end of it.    $m \cdot N$

  A teenager pushes tangentially on a small hand-driv rs m7ysmo. w;(gbb45hen merry-go-round and is able to accelerate it from rest to a frequency of 15 rpm in 10.0 s. Assume the merry-go-round is a uniform disk of radius 2.5 m and has a mass of 760 kg, and two children (each with a mass of 25 kg) sit opposite each other os7byr5m m4w.b ( ;gshon the edge. Calculate the torque required to produce the acceleration, neglecting frictional torque. $\approx$   $m \cdot N$ What force is required at the edge?    N

  A centrifuge rotor rotating at 10,300 rpm is shut off and is eventua0*0n3l;x i4l8hn m aziyy uob+lly brought 3b*+o 4mi hxn 8yuzla 0lyni;0uniformly to rest by a frictional torque of 1.2 $m \cdot N$ If the mass of the rotor is 4.80 kg and it can be approximated as a solid cylinder of radius 0.0710 m, through how many revolutions will the rotor turn before coming to rest,    $rev$ how long will it take?    s

  The forearm in Fig. 8mq5rgx 7:fo-6 8er dnl–45 accelerates a 3.6-kg ball at 7 $m/s^2$ by means of the triceps muscle, as shown. Calculate
(a) the torque needed,    $m \cdot N$
(b) the force that must be exerted by the triceps muscle. Ignore the mass of the arm.    N

  Assume that a 1.00-kg ball i4w) ig.) wzp-c/i4 8hyngfl ans thrown solely by the action of the forearm, which rcgi -af)/i8wnn.hz4 4 ) pwylgotates about the elbow joint under the action of the triceps muscle, Fig. 8–45. The ball is accelerated uniformly from rest to 10 $m/s$ in 0.350 s, at which point it is released. Calculate
(a) the angular acceleration of the arm,    $rad/s^2$
(b) the force required of the triceps muscle. Assume that the forearm has a mass of 3.70 kg and rotates like a uniform rod about an axis at its end.    N

  A helicopter rotor blade can be considered a long thin rod, as shown in Figzkv+ i3(ofp1l . 8–46.opk+vz3 fli1(
(a) If each of the three rotor helicopter blades is 3.75 m long and has a mass of 160 kg, calculate the moment of inertia of the three rotor blades about the axis of rotation.    $kg \cdot m^2$
(b) How much torque must the motor apply to bring the blades up to a speed of 5 $rev/s$ in 8.0 s?    $m \cdot N$

An Atwood’s machine considyvoo l5 h1 j.3y gc94lpdx /yw4o.j)dsts of two masses, $m_1$ and $m_2$ which are connected by a massless inelastic cord that passes over a pulley, Fig. 8–47. If the pulley has radius R and moment of inertia I about its axle, determine the acceleration of the masses $m_1$ and $m_2$ and compare to the situation in which the moment of inertia of the pulley is ignored. [Hint: The tensions $F_{T1}$ and $F_{T2}$ are not equal. We discussed this situation in Example 4–13, assuming for the pulley.]

  A hammer thrower accelerates the hammer frn1f90qu b)7ccn.8p bt zuef 0qom rest within four full turns (revolutions) and releases it at a speed 01b7 fz).b8cnfeqq0 t c9uunp of 28 $m/s$ Assuming a uniform rate of increase in angular velocity and a horizontal circular path of radius 1.20 m, calculate
(a) the angular acceleration,    $rad/s^2$
(b) the (linear) tangential acceleration,    $m/s^2$
(c) the centripetal acceleration just before release,    $m/s^2$
(d) the net force being exerted on the hammer by the athlete just before release,    N
(e) the angle of this force with respect to the radius of the circular motion.    $^{\circ} $

  A centrifuge rotor has a mommt: r:o32 kylxent of inertia of $3.75 \times10^{-2 }$ $kg \cdot m^2$ How much energy is required to bring it from rest to 8250 rpm?    J

  An automobile engine develops a torque of 28 wj0 rj:w 9tcfv.z5/fc0 $m \cdot N$ at 3800 rpm. What is the power in watts and in horsepower?    W    hp

  A bowling ball of mai zk q;ir3(rlrw2qgm-0i:7 iqj.kgp -ss 7.3 kg and radius 9.0 cm rolls without slipping down a lane at 3krlqgimqii ;3- (0k w7jg zp2r-ri:. q.3 $m/s$ Calculate its total kinetic energy.    J

  Estimate the kinetic energy oftj/ pc3y .1qjo the Earth with respect to the Sun as the sum of two terms.3/ ty ocq1jjp,
(a) that due to its daily rotation about its axis,$KE_{daily}$=    $\times10^{29 }$ J
(b) that due to its yearly revolution about the Sun. $KE_{yearly}$+    $\times10^{33 }$ J [Assume the Earth is a uniform sphere with $6 \times10^{ 24}$ kg and $6.4 \times10^{6 }$ m and is $1.5 \times10^{8 }$ km from the Sun.]$KE_{daily}$ + $KE_{yearly}$ =    $ \times10^{33 }$ J

  A merry-go-round has a mass of 1640 kg and a radius of 7.50 m. How muc7a mcapj.t;g da7xdj/m; u(7gcr4 j0dh net work is required to accelerate .atpc 4ram djgg7dc7a/0 u7dm;( jx;jit from rest to a rotation rate of 1.00 revolution per 8.00 s? Assume it is a solid cylinder.    J

  A sphere of radius 20.0 cm and mass 1.80 kg start/jray/6 gbyn11rhmrfaf0: uz3 1 hrz6s from rest and rolls without slipping r3/f: 6r161 fh0aruzg b yyzran/jmh1 down a 30.0 $^{\circ} $ incline that is 10.0 m long.
(a) Calculate its translational and rotational speeds when it reaches the bottom. $v_{CM}$ =    $\omega$ =    $rad/s$
(b) What is the ratio of translational to rotational KE at the bottom?    Avoid putting in numbers until the end so you can answer:
(c) do your answers in (a) and (b) depend on the radius of the sphere or its mass?

  Two masses, $m_1$ = 18 kg and $m_2$ = 26.5 kg are connected by a rope that hangs over a pulley (as in Fig. 8–47). The pulley is a uniform cylinder of radius 0.260 m and mass 7.50 kg. Initially, is on the ground and $m_2$ rests 3.00 m above the ground. If the system is now released, use conservation of energy to determine the speed of $m_2$ just before it strikes the ground. Assume the pulley is frictionless.    $m/s$

  A 2.30-m-long pole is)t0 eoc)j bc e6o::shv balanced vertically on its tip. It starts to fall and its lower end does not slip. What will be the speed of the upper end of the pole jush):: ot be )s0vjeocc6t before it hits the ground? [Hint: Use conservation of energy.]    $m/s$

  What is the angular momentum of a 0.210-kg ball rotating on the end of a5lg1i9 uy*hw s thin string 5uislg91wy*h in a circle of radius 1.10 m at an angular speed of 10.4 $rad/s$?    $kg \cdot m^2$

  (a) What is the angular momentum of a 2.8-kg uniform cylindrical grinding wheel +-(arkkzu dz( of radius 18 cm when rotating at 15d(+az-rzuk (k00 rpm?    $kg \cdot m^2$
(b) How much torque is required to stop it in 6.0 s?    $m \cdot N$

A person stands, hands at his 0yxxgrg3ic 92tv.1 yu +yt3 yjside, on a platform that is rotating at a rate of 1.3rev/s I +yytgiv1uxrxj.y 3039ygtc 2f he raises his arms to a horizontal position, Fig. 8–48, the speed of rotation decreases to 0.8 $rev/s$ (a) Why?
(b) By what factor has his moment of inertia changed?

  A diver (such as the one shown in Fig. 8–29) can reduce heripb9r/sc 0qb4q8i0m l moment of inertia by a factor of about 3.5 when changing from the straight position to the tuck position. If she makes 2.0 rotations in 1.5 s whe sbi8 0 bl04q9c/qpmrin in the tuck position, what is her angular speed ($rev/s$) when in the straight position?   $rev/s$

  A figure skater can increase her spin rotation rate from an init 1de4a6*q8 ;ywv*ga.dd wo cssial rate of 1.0 rev every g.;*ao*ds8d w a1se c6yd4qvw2.0 s to a final rate of 3 $rev/s$ If her initial moment of inertia was 4.6 kg*$m^2$ what is her final moment of inertia? How does she physically accomplish this change?    $kg \cdot m^2$

  A potter’s wheel is rotating around a verticaloo fe:d0,y pm -x blyy:y049xt axis through its center at a frequency of 1.5rev/s The wheel can be considered a uniform disk of mass 5.0 kg and diameter 0.40 m. The potter then throws a 3.1-kg chunk of clay, approximately shaped as a9 yeolpm,f0x-x :ot 0y4yby:d flat disk of radius 8.0 cm, onto the center of the rotating wheel. What is the frequency of the wheel after the clay sticks to it?    $rev/s$

  (a) What is the angular momentum of a figure skater spi pvkx5/4unm)7 0srs1 jdchz8 bfymq9pd 1.yb4nning at 3.5 $rev/s$ with arms in close to her body, assuming her to be a uniform cylinder with a height of 1.5 m, a radius of 15 cm, and a mass of 55 kg?    $kg \cdot m^2$
(b) How much torque is required to slow her to a stop in 5.0 s, assuming she does not move her arms?    $m \cdot N$

  Determine the angular mo8m + 2i h7fa,vrje*xp; lbmezv0(2ncfmentum of the Earth
(a) about its rotation axis (assume the Earth is a uniform sphere),    $\times 10^{33} \; kg \cdot m^2$
(b) in its orbit around the Sun (treat the Earth as a particle orbiting the Sun). The Earth has mass $6 \times 10^{24} \; kg$ and radius $6.4 \times 10^{6} \; m$ and is $1.5 \times 10^{8} \; km$ from the Sun.    $\times10^{40} \; kg \cdot m^2$

A nonrotating cylindrical disk of moment of inertia I is dropped onto an identi i,z+/ ,xjuswpcal disk rotating at angular spee p/jius,zw,x +d $\omega$ Assuming no external torques, what is the final common angular speed of the two disks?

  A uniform disk turnsl2;( 8 ti5/qgsmq9fusii:n7 tr um8kc at 2.4 $rev/s$ around a frictionless spindle. A nonrotating rod, of the same mass as the disk and length equal to the disk’s diameter, is dropped onto the freely spinning disk, Fig. 8–49. They then both turn around the spindle with their centers superposed. What is the angular frequency in rev/s of the combination?    $rev/s$

  A person of mass 75 kg stands at the center ofawyp,rgu 4y 4- cnn(v/ a rotating merry-go-round platform of radius 3rv44-n w ,y(ucnga py/.0 m and moment of inertia 920 $kg \cdot m^2$ The platform rotates without friction with angular velocity 2 $rad/s$ The person walks radially to the edge of the platform.
(a) Calculate the angular velocity when the person reaches the edge.    $rad/s$
(b) Calculate the rotational kinetic energy of the system of platform plus person before and after the person’s walk.$KE_i$ =    J $KE_f$ =    J

  A 4.2-m-diameter merry-go-round is rotating freely with an angular velv sn)i46w,gxoocity of 0.8 xwg ,oni64vs )$rad/s$ Its total moment of inertia is 1760 $kg \cdot m^2$ Four people standing on the ground, each of mass 65 kg, suddenly step onto the edge of the merry-go-round. What is the angular velocity of the merry-go-round now?    $rad/s$ What if the people were on it initially and then jumped off in a radial direction (relative to the merry-go-round)?    $rad/s$

  Suppose our Sun eventu1lqd,vn /kj *1cscr2: exj/qvally collapses into a white dwarf, losing about half its mass in the process, and wind1qv12ln,/q*rvcx ke j/s jcd:ing up with a radius 1.0% of its existing radius. Assuming the lost mass carries away no angular momentum, what would the Sun’s new rotation rate be?(round to the nearest integer)$\approx$    $rad/s$ (Take the Sun’s current period to be about 30 days.) What would be its final KE in terms of its initial KE of today?$KE_{f}$=    $KE_{i}$

  Hurricanes can involve winds in exce 9vq,ywjc- ;po il47buss of 120 $km/h$ at the outer edge. Make a crude estimate of
(a) the energy,    $ \times10^{16 }$ J
(b) the angular momentum, of such a hurricane, approximating it as a rigidly rotating uniform cylinder of air (density 1.3 $kg \cdot m^2$) of radius 100 km and height 4.0 km.    $ \times10^{20 }$ $kg \cdot m^2$

  An asteroid of mass hc2ig/;qpdvmw 2f+e+$ 1.0\times10^{ 5}$ traveling at a speed of relative to the Earth, hits the Earth at the equator tangentially, and in the direction of Earth’s rotation. Use angular momentum to estimate the percent change in the angular speed of the Earth as a result of the collision.    $\times10^{-16 }$ %

A person stands on a platform, initp z cv)p m l22-9-hcqjyzjb5.oially at rest, that can rotate freely without friction. The moment of inertia of the peo5zjc-ph 9 lmjzycv2. p-)2 bqrson plus the platform is $I_P$ The person holds a spinning bicycle wheel with its axis horizontal. The wheel has moment of inertia $I_W$ and angular velocity $\omega_W$ What will be the angular velocity $\omega_W$ of the platform if the person moves the axis of the wheel so that it points (a) vertically upward, (b) at a 60º angle to the vertical, (c) vertically downward? (d) What will $\omega_P$ be if the person reaches up and stops the wheel in part (a)?

  Suppose a 55-kg person stands at the edge of a 6.5-m diameter merr0mq )tba k6lp9y-go-round turntable that is mounted on frictionless bearings and has a moment of inertia of 1700 0lamk9p 6b)tq $kg \cdot m^2$ The turntable is at rest initially, but when the person begins running at a speed of 3.8 $m/s$ (with respect to the turntable) around its edge, the turntable begins to rotate in the opposite direction. Calculate the angular velocity of the turntable.    $rad/s$

A large spool of rope rolls on the ground with the end of the rope lyin3+j lixwhtu z/obu5.7 g on the top edge of the spool. A person grabs the end of the rope and walks a distance L, holding onto it, Fig. 8–50. The spool rolls behind tutl/xo37+5 .h uibjwzhe person without slipping. What length of rope unwinds from the spool? How far does the spool’s center of mass move?

  The Moon orbits the E,5,p +ynr+:c/vnw9urb ih1t mf ma lo:arth such that the same side always faces the Earth. Determine the ratio of the Moon’s spin angular momentum (about its own axis) to its orbital angular momentum. (In the lattocin+ +lf1,, vrhmb9r:uyn pt5/:aw mer case, treat the Moon as a particle orbiting the Earth.)    $\times10^{ -6}$

  A cyclist accelerates from rest fd5 o(oh4v5 ;x jedv-s:mjh1p at a rate of 1 m/$s^2$ How fast will a point on the rim of the tire at the top be moving after 3.0 s? [Hint: At any moment, the lowest point on the tire is in contact with the ground and is at rest — see Fig. 8–51.]    $m/s$

  A 1.4-kg grindstone in the shape of a uniform cylinder of radius bblhk m kq7*o+9n9:wvkn:p ; p0.20 m acquires a rotational rate of from rest over a 6.0-s interval at constant angnbvnk q;9:* lw9kpp+om hb:7kular acceleration. Calculate the torque delivered by the motor.    $m \cdot N$

  (a) A yo-yo is made of two solid cylindrical dislxd+rk.r6ossi)oc 7bf5i mbzx 86:d 2ks, each of mass 0.050 kg and diameter 0.075 m, joined by a (concen56msfxick b zr8:o 7.r6i)x2+ lsdobdtric) thin solid cylindrical hub of mass 0.0050 kg and diameter 0.010 m. Use conservation of energy to calculate the linear speed of the yo-yo when it reaches the end of its 1.0-m-long string, if it is released from rest.    $m/s$
(b) What fraction of its kinetic energy is rotational?    %

  (a) For a bicycle, how is the angular sph,fuoja tj :8-eed of the rear wheel ($\omega_R$) related to that of the pedals and front sprocket ($\omega_F$) Fig. 8–52? That is, derive a formula for ($\omega_R$)/($\omega_F$) Let $N_F$ and $N_R$ be the number of teeth on the front and rear sprockets, respectively. The teeth are spaced equally on all sprockets so that the chain meshes properly.
(b) Evaluate the ratio ($\omega_R$)/($\omega_F$) when the front and rear sprockets have 52 and 13 teeth, respectively,   
(c) when they have 42 and 28 teeth.   

  Suppose a star the size of our Sun, but with mass 8.0 times as grffb r9;.8dyszeat, were rotating at a speed of 1.0 revolution every 12 days. If it were to undergo gravitational collapse to a neutron star of radius 11 ;s9.dyfb8f z rkm, losing three-quarters of its mass in the process, what would its rotation speed be? Assume that the star is a uniform sphere at all times, and that the lost mass carries off no angular momentum.    $\times10^{9 }$ $rev/day$

  One possibility for a low-pollution automobile is for it to use yxg)+.py1 m kx;tb u3q,(i kgyenergy stored in a heavy rotating flywheel. Suppose such a car has a total mass of 1400 kg, uses a unifo,13g kku (x m.+; bxgqyiytp)yrm cylindrical flywheel of diameter 1.50 m and mass 240 kg, and should be able to travel 350 km without needing a flywheel “spinup.”
(a) Make reasonable assumptions (average frictional retarding force = 450N twenty acceleration periods from rest to equal uphill and downhill, and that energy can be put back into the flywheel as the car goes downhill), and show that the total energy needed to be stored in the flywheel is about $ 1.7\times10^{8 }$J.    $ \times10^{ 8}$ J
(b) What is the angular velocity of the flywheel when it has a full “energy charge”?    $rad/s$
(c) About how long would it take a 150-hp motor to give the flywheel a full energy charge before a trip? $\approx$    min

  Figure 8–53 illustrates an tqbil 5+8s*3k/ylpqd$H_2O$ molecule. The O–H bond length is 0.96 nm and the H–O–H bonds make an angle of 104 $^{\circ} $. Calculate the moment of inertia for the $H_2O$ molecule about an axis passing through the center of the oxygen atom
(a) perpendicular to the plane of the molecule,    $\times10^{-45 }$ $kg \cdot m^2$
(b) in the plane of the molecule, bisecting the H–O–H bonds.    $ \times10^{-45 }$ $kg \cdot m^2$

  A hollow cylinder (hoop) isiwpqp )s-:fa4o4 i pb1 rolling on a horizontal surface at speed v=3.3 $m/s$ when it reaches a 15 $^{\circ} $ incline.
(a) How far up the incline will it go? $\approx$    m (round to one decimal place)
(b) How long will it be on the incline before it arrives back at the bottom?    s

A uniform rod of mass M and length L can pivot freely (i.e., we avj;42cuka :x *s mul51b8unw8p0jbcignore friction) about a hinge attached to a wall, as in Fig. 8–54. The rod is held horizontally and then released. At the moment of release, determine (a) the angular acceleration of the rod, and (b) the linear acceleratio;kx c1mc uslaj:485 uv2ub0ap*b8 n jwn of the tip of the rod. Assume that the force of gravity acts at the center of mass of the rod, as shown. [Hint: See Fig. 8–21g.]

A wheel of mass M has radiusz0b4fi:q;9yam+c2mo paht 9 m R. It is standing vertically on the floor, and we want to exert a horizontal force F at its axle so that it will climb a step against wh 04a my+p;9imtq2mb9h fc:ao zich it rests (Fig. 8–55). The step has height h, where h

  A bicyclist traveling with speed v=4.2m/s on a flat road is mvg8(ub 9 yf.hxaking a turn with a radius The forces acg(8ufb v9xy .hting on the cyclist and cycle are the normal force $\left(\mathbf{\vec{F}}_{\mathrm{N}}\right)$ and friction force $\left(\mathbf{\vec{F}}_{\mathbf{fr}}\right)$ exerted by the road on the tires, and $m\vec{\mathbf{g}}$ the total weight of the cyclist and cycle (see Fig. 8–56).
(a) Explain carefully why the angle $\theta$ the bicycle makes with the vertical (Fig. 8–56) must be given by tan $\tan\theta=F_{\mathrm{fr}}/F_{\mathrm{N}}$ if the cyclist is to maintain balance.(round to the nearest integer)
(b) Calculate $\theta$ for the values given.    $^{\circ} $
(c) If the coefficient of static friction between tires and road is $\mu_s=0.70$ what is the minimum turning radius?    m

  Suppose David puts a 0.50-kg rock into a sling of length 1.5 m p4oy un;* 8rmband begins whirling the rock in a nearly horizontal circle above his head, accelerating it from rest to a rate of 120 rpm after 5.0 s. What is the torque required to achieve this feat, anrybmu;4o *8npd where does the torque come from?    $m \cdot N$

  Model a figure skater’s body as a solid cylinder gq y6htqy6tyk:(d/ko hzt;t riat-(2t y 2q /9and her arms as thin rods, making reasonable estimates for the dikh-(iky69 g2q 2y/(;yzr d6ho :qty / tqtatttmensions. Then calculate the ratio of the angular speeds for a spinning skater with outstretched arms, and with arms held tightly against her body.   

  You are designing a clutch assembly which consists of two cylindrical pltzmj4 o; 8w8xa)2wr dkigb 9i7 kp5,rtates, of 452zg k)ib8k7 ia;rrpw mt,odw8x9 jt mass $M_{\mathrm{A}}=6.0$ $\mathrm{kg}$ and $M_{\mathrm{B}}=9.0$ $\mathrm{kg}$ with equal radii R=0.60 $\mathrm{m}$ They are initially separated (Fig. 8–57). Plate $M_{\mathrm{A}}$ is accelerated from rest to an angular velocity $\omega_1=7.2$ $\mathrm{rad/s}$ in time $\Delta t=2.0$ s Calculate
(a) the angular momentum of $M_{\mathrm{A}}$    $kg \cdot m^2$
(b) the torque required to have accelerated $M_{\mathrm{A}}$ from rest to $\omega_{1}$    $m \cdot N$
(c) Plate $M_{\mathrm{B}}$ initially at rest but free to rotate without friction, is allowed to fall vertically (or pushed by a spring), so it is in firm contact with plate $M_{\mathrm{A}}$ (their contact surfaces are high-friction). Before contact, $M_{\mathrm{A}}$ was rotating at constant $\omega_{1}$ After contact, at what constant angular velocity $\omega_{s}$ do the two plates rotate?    $rad/s$

  A marble of mass m and radius r rolls along the loopek;vq khe )cpjje0ji*n 3- .wp7d rough track of Fig. 8–58. What is the minimum value of the vertical height h that the marble must drop if it is to reach the high3j7.kp;ije0n- )q jvcew phk* est point of the loop without leaving the track? Assume $r\ll R$ and ignore frictional losses. h =    R

  Repeat Problem 84, but do not asscx3j rp,qpjgi2 .y rjc v-638jume $r\ll R$ h =    (R-r)

  The tires of a car make 85 revolutions as the car reduces its speed uniformly y1o7 u97hcbzv from 90km/h to 60km/h The tires have a diameter of 0.90 m. (a) What was the angular accelerac7y1 ubh9z 7ovtion of each tire? $\approx$    $rad/s^2$(round to two decimal place)
(b) If the car continues to decelerate at this rate, how much more time is required for it to stop?    s